The linear determinant multiplicity conjecture for two-dimensional topological theories

Let $s\ne 0$ be an integer, and let $\alpha(s)=(\beta_0,s,0,0,\dots)$, so that the generating function is $Z_{\alpha(s)}(T)=\beta_0+sT$. For arbitrary $\beta_0$ in characteristic zero, let $\mathrm{det}_n$ be the determinant polynomial and let $A_{\alpha(s)}(n)$ be the corresponding quotient space. Linear determinant multiplicity conjecture. The exponent of the factor $\beta_1-s$ in $\mathrm{det}_n$ is

$$a_n-\dim A_{\alpha(s)}(n).$$

The conjecture predicts the multiplicity of each nonzero integral root of the determinant polynomial and is supported by the general lower bound arising from the null space of the associated bilinear form; the source does not state a resolution.