Widom–Rowlinson partition-function conjecture for distinct activities

Let $G$ be a $d$-regular graph. For positive activities $\lambda_1,\lambda_2$, define the distinct-activity Widom–Rowlinson partition function by

$$P_G(\lambda_1,\lambda_2)=\sum_{\chi\in\Omega(G)}\lambda_1^{X_1(\chi)}\lambda_2^{X_2(\chi)}.$$

Then

Distinct-activity partition-function conjecture. For any $\lambda_1,\lambda_2>0$,

$$P_G(\lambda_1,\lambda_2)\leq P_{K_{d+1}}(\lambda_1,\lambda_2)^{|V(G)|/(d+1)}.$$

This asserts that the complete graph $K_{d+1}$ maximises the normalised partition function among all $d$-regular graphs. The supplied material gives no resolution or status evidence beyond the conjecture statement.