Growth monotonicity of Q-Kostka polynomials under adjoining a part

Let $\lambda$ and $\mu$ be partitions, let $r$ be an integer that is not a part of either partition, and let $L_{\lambda\mu}(q)$ denote the $Q$-Kostka polynomial. Growth conjecture. One has

$$L_{\lambda+(r),\,\mu+(r)}(q)\geq L_{\lambda\mu}(q).$$

This is proposed as an analog of a growth property of the Kostka–Foulkes polynomials, attributed in the paper to Gupta. The source presents it as a conjectural property and does not give a general proof.