The interchange-process comparison inequality

Let $k\geq 2$, let $\mathcal S_k$ be the symmetric group on $k$ letters, and let $g:\mathcal S_k\to\mathbb R$ be any function. For nonnegative numbers $\gamma_1,\ldots,\gamma_{k-1}$, with an empty sum interpreted as zero and $(ij)\in\mathcal S_k$ denoting the transposition of $i$ and $j$, the interchange-process comparison conjecture.

$$\sum_{\sigma\in\mathcal S_k}\sum_{i=1}^{k-1}\gamma_i\bigl[g(\sigma)-g((ik)\sigma)\bigr]^2\geq\sum_{\sigma\in\mathcal S_k}\sum_{1\leq i<j\leq k-1}\frac{\gamma_i\gamma_j}{\gamma_1+\cdots+\gamma_{k-1}}\bigl[g(\sigma)-g((ij)\sigma)\bigr]^2.$$

This inequality was proposed as a tool for controlling spectral changes in an induction argument for Aldous' conjecture, when a new vertex has degree $k-1$. The supplied context describes it as a conjecture and says that the broader Aldous conjecture had been resolved, but gives no resolution status for this inequality itself.