Product-type conjecture for one-cycle structure constants

Let $q$ be a prime power, let $\Phi$ be the set of monic irreducible polynomials over $\mathbb F_q$ other than $t$, and let $[d]$ denote the $q$-integer. Fix $\xi\in\mathbb F_q^*$ and let $\boldsymbol{\lambda}=(1)_{t-\xi}$. Let $f'\in\Phi$ have degree $d-1$ and constant term $a$, and set $\boldsymbol{\mu}=(1)_{f'}$.

One-cycle product conjecture. (1) For every irreducible polynomial $f\in\mathbb F_q[t]$ of degree $d$ with constant term $-\xi a$, there exist $g,h\in G_d$ of modified types $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$, respectively, such that the modified type of $gh$ is $(1)_f$. (2) If $f\in\Phi$ has degree $d\geq 3$ and constant term $b=-\xi a$, then

$$a^{(1)_f}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}=[d].$$

The first part is known when $\|\boldsymbol{\mu}\|=1$, while the second is supported in lower-degree cases and by examples. The general assertions remain unresolved in the source.