Conjecture on the recursive coefficients in Boij–Söderberg decompositions

Let $a_1,\ldots,a_{c+1}$ be the parameters and let $a$, $r_1,\ldots,r_\varepsilon$, $z_1,\ldots,z_\varepsilon$, $y_s$, $n$, and $\varepsilon$ have the meanings from Theorem~. Assume

$$a_{c+1} > \max\left\{a,\frac{r_1}{z_1},\ldots,\frac{r_\varepsilon}{z_\varepsilon}\right\}.$$

Recursive-coefficient conjecture. For $\varepsilon<s<\varepsilon+c$, one has

$$y_s=c! \cdot a_1\cdots a_c \cdot \left(a_{c+1}-\sum_{i=1}^{s-n}(a_{c+1-i}-a_i)\right).$$

The conjecture gives an explicit formula for the coefficients in the second phase of the recursive algorithm. The paper reports that it holds in codimension at most $3$, while the general case remains open.