The following conjectures from Kapranov and Saito's *Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions* aren't as well-known as they aught to be, so I'd like to state them and then ask if any progress has been made since the paper was released in 1997.

I quote (more-or-less):

Fix a ring $A$. By a **hieroglyph** we will mean an oriented graph $\Gamma$ without oriented loops, equipped with the following additional structure:
(a) An assignment of a positive integer to each vertex of $\Gamma$ so that all these integers are distinct.
(b) An assignment of a nonempty ordered sequence of elements of $A$ to each edge of $\Gamma$.

The number of elements written on the edge of a hieroglyph is called the **weight** of the
edge and the weight of the whole hieroglyph is by definition the sum of
weights of all the edges.

ConjectureFor every hieroglyph $\Gamma$ there is a polyhedral ball $P (\Gamma)$ with the following properties:

$(a)$ The dimension of $P (\Gamma)$ is equal to the weight of $\Gamma$. The combinatorial type of $P (\Gamma)$ depends only on the underlying graph of $\Gamma$ and on weights of the edges.

$(b)$ If $\Gamma = \cup \Gamma_i$ is the irreducible decomposition of a hieroglyph $\Gamma$, then $P (\Gamma) = \Pi P (\Gamma_i)$.

$(c)$ The boundary of each $P (\Gamma)$ is composed of the balls $P (\Gamma)'$ for some hieroglyphs $\Gamma'$.

$(d)$ Let $B$ be the union of the polyhedral balls $P (\Gamma)$ for all the hieroglyphs $\Gamma$ according to the identifications of their boundaries given by part $(c)$ above.
Then $B$ is the homotopy ﬁber of the natural map $BGL(A) \to BGL^+(A)$. Another model for this homotopy fiber is given by **Volodin K-theory** in terms of the classifying space of the groups of upper triangular matrices in $GL(A)$. So a proof here would give a "polyhedral model" for this K-theory space.

$(e)$ For the hieroglyph of Dynkin type $A_n$, $P(A_n)$ is the **associahedron** (Stasheff polytope) $\cal K_{n+1}$....this should be doable, but I have not seen a proof in the literature.

This is wonderful stuff - does anyone know how to verify any of these claims? Or are they open?

The paper goes on to elucidate many connections of hieroglyphs to Morse theory and algebraic K-theory, so solutions should be very interesting. Is anyone working on this kind of thing? The constructions in this paper are very concrete, and I'd like to think that a lot more can be said.