(This question is tangentially related to an earlier
question I posed: [Minimal surface enclosing two congruent balls](https://mathoverflow.net/q/410494/6094).)

Let $B_1,\ldots,B_k$ be unit-radius balls in $\mathbb{R}^3$, with pairwise
disjoint interiors.
Let $S$ be the minimal area surface that (a) encloses $B_1,\ldots,B_k$
and (b) is topologically a sphere.
Then:
> ***Q***. Is it the case that never is $S$ the convex hull of $B_1,\ldots,B_k$?

It may be difficult to specify the exact structure of the minimal surface,
but can we at least prove that it always "dents inward,"
and so is never the convex hull?

The "is topologically a sphere" condition requires the balls to be
relatively closely packed. Otherwise $S$ might fracture into several components.

This image from
[Wrapping juggling balls](https://mathoverflow.net/q/349048/6094) illustrates the convex hull of three congruent balls:

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<img src="https://i.sstatic.net/PZFDY.jpg" />

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