Singularity-free isotopies between string diagrams for monoidal categories Given a monoidal category, it is a theorem of Joyal and Street that an equation between string diagrams is provable from the axioms if and only if there is a recumbent isotopy that relates them. The recumbent property requires that the family of critical points of the string diagram (with respect to the height coordinate) do not change during the isotopy. We also require that the isotopy does not move the boundary points.
It seems a reasonable conjecture that this recumbent property is unnecessary: that isotopic string diagrams are always related by a recumbent isotopy (this conjecture is made in print by Selinger; see page 11 of A survey of graphical languages for monoidal categories.)
For example, consider the following pair of isotopies of string diagrams. The lower is recumbent, but the upper is not, since it introduces additional intermediate critical points.

Do isotopic string diagrams always admit a recumbent isotopy, as in this case?
As an extension, one could also consider whether the recumbent property can be removed in the definition of isotopy for braided and symmetric monoidal categories.
 A: If I understand your question correctly, it seems that there is a problem of winding numbers.  Your diagrams seem to allow ending on univalent vertices.  Consider, then, a string that starts at a univalent vertex, winds around that vertex a few times, and then heads off.  It seems that non-recumbent isotopies allow that diagram to unwind, whereas recumbent ones do not.  Since pivotality is extra structure on a rigid category, this almost surely matters.
Here I will try to draw in ASCII the non-recumbent-isotopic diagrams I have in mind:
  |      ?       _  \
  |      =      / \  \
  °            |   °  |
                \____/

But my suspicion is that some version of "winding number" is the only extra thing you need to track.

Edit: In the comments below, Jamie explained how I had misinterpreted the question.  I'll leave my answer above so that those comments make sense, and now try to give an answer to the clarified question, namely:
I believe the answer to your question is "yes, isotopic string diagrams are recumbent-isotopic."  I will try to outline a proof, but I have not thought through carefully all details, so this is merely an "idea of a proof."  In the argument will come up one subtlety, suggesting that even though the answer is "yes", it is so in an imperfect way.  In particular, I really haven't thought about whether this extends to braided (with or without a twisting/ribbon/SO(2) structure) or symmetric categories.
Let me recall the rules.  Diagrams are drawn in a disk, and isotopies must fix the boundary of the disk pointwise.  Then the diagram consists of some one- and zero-dimensional strata, and also the 2-dimensional strata left over from removing the actual drawing.
Rather than working with a height function, let me work with a framing, since I think better in that language.  The first direction of the framing will be in the direction of increasing height; the second will be 90° counterclockwise of that.  The rule is that edges must follow the first direction, and there's a similar rule at vertices.  "Recumbent" isotopies are isotopies of the framed diagram.
(Aside: come to think of it, is it obvious that working with height functions and recumbent isotopies is "the same" as working with framings?  There's some issue with how the two versions treat the 2-dimensional strata.)
Now let's say I have an isotopy of the disk (with diagram drawn in it) as an unframed object, and I frame the starting position.  I can drag that framing through the isotopy, thereby framing the final position.  I assume the initial framing follows the rules; then so will, of course, the dragged-through framing.  The question is to compare the dragged-through framing with some other rule-following framing of the same final position.  Are these two framed diagrams framed-isotopic?
Well, we have the same diagram with two rule-following framings.  How different can they be?  Along the 0- and 1-dimensional strata they must agree, because the framings there are forced by the rules.  What about in the 2-dimensional strata?
Each two-dimensional stratum is a connected domain in the disk.  There aren't very many of these: they are just disks with some punctures.  Consider first the case with no punctures.  Then we have a disk with its boundary framed in the unique way so that its framing extends into the interior of the disk.  I believe that such an extension is homotopically unique.  (Is this obvious?)
When the subtlety alluded to earlier arises is if there are punctures in the stratum, i.e. when a 2-dimensional stratum is not contractible.  This arises exactly when your diagram includes some endomorphisms of the (invisible) unit object.  Then you have a disk with some holes in it, and a framing of the boundary.  Now the extension to the interior is not unique.
But I think that different extensions can't be that different.  I mean, I think there is an isotopy relating them.  This should be the same as the assertion that the space of k-ary operations in the "framed" version of the E_2 operad is connected.
So in summary, I think that the only difference between recumbent and non-recumbent isotopies is that the latter allow taking an endomorphism of the unit object and "spinning it around".
 •        •                °                 •
 |  ->   /    ->  °-•  ->   \    -> .... ->  |
 °      °                    •               °
    This is not the identity isotopy.

 •       /\         /\       •   /\       •     °      •          •
 |  ->  |  •  ->   |  |  ->  |  |  |  ->  |     |  ->  |      ->  |
 °      °          °  /       \ °  /       \    /       \         °
                     /         \  /         \  /         \  °
                    •           \/           \/           \/
    Another drawing of the same isotopy, using moves like in your picture.

A: This conjecture was confirmed recently, with two different methods:

*

*By Jamie and I, as a corollary of our work on the word problem for monoidal categories (in which we give algorithms to detect whether two diagrams are equivalent up to the axioms of monoidal categories): arXiv:1804.07832


*By Xuexing Lu, who proved it with a more direct topological argument that I am still to digest fully: arXiv:2010.11582
