Kerov's stochastic monotonicity conjecture for Jordan types over finite fields

Let $Y_n$ denote the set of Young diagrams with $n$ boxes. Let $\lambda,\hat\lambda\in Y_n$ and $\mu,\hat\mu\in Y_{n-1}$ be pairs of Young diagrams such that both pairs differ by moving the box $(i,j)$ to $(\hat i,\hat j)$, where $\hat i>i$, and suppose that

$$\lambda\setminus\mu=\hat\lambda\setminus\hat\mu=(r,c).$$

Kerov's stochastic monotonicity conjecture. If $r<i$, then

$$\frac{\dim_t(\hat\mu\nearrow\hat\lambda)}{\dim_t(\hat\lambda)}\geq\frac{\dim_t(\mu\nearrow\lambda)}{\dim_t(\lambda)}.$$

If $r>\hat i$, then

$$\frac{\dim_t(\hat\mu\nearrow\hat\lambda)}{\dim_t(\hat\lambda)}\leq\frac{\dim_t(\mu\nearrow\lambda)}{\dim_t(\lambda)}.$$

Here $\dim_t(\lambda)$ counts uni-uppertriangular matrices over $\mathbb F_p$ whose Jordan diagram is $\lambda$, and $\dim_t(\mu\nearrow\lambda)$ counts those whose top-left $(n-1)\times(n-1)$ corner has diagram $\mu$ and whose diagram is $\lambda$. This is a proposed monotonicity property for the transition probabilities in the Hall--Littlewood deformation of the Young graph; the source presents it as the inequality to be proved in the case $t=p^{-1}$, and no resolution is supplied.