Log-concavity conjecture for parabolic Kostka numbers

Let $\lambda$ be a partition and let $R=((m_1^{l_1}), \dots, (m_k^{l_k}))$ be a sequence of rectangular partitions. For an integer $N \geq 1$, define

$$NR=(((Nm_1)^{l_1}),\dots,((Nm_k)^{l_k})).$$

Let $K_{\lambda,R}$ denote the parabolic Kostka number associated to $\lambda$ and $R$. Log-concavity conjecture for parabolic Kostka numbers. Let $\lambda$ be a partition and $R$ be a sequence of rectangular partitions. Then

$$K_{(N+1)\lambda,(N+1)R} \cdot K_{(N-1)\lambda,(N-1)R} \leq (K_{N\lambda,N R})^2,$$

for every integer $N \geq 1$. This is a proposed extension of the log-concavity phenomenon to parabolic Kostka numbers; the supplied source does not provide evidence resolving the conjecture in general.