Conjecture on the induced forest number of Paley graphs of square order

Let $q>7$ be a prime power, and let $\mathcal{P}(q^2)$ denote the Paley graph on the finite field of order $q^2$. Let $\tau(G)$ denote the maximum order of an induced forest in a graph $G$. Induced forest conjecture. For $q>7$ a prime power,

$$\tau(\mathcal{P}(q^2))=q+1.$$

The preceding computational searches found that adding two vertices to a maximum independent set does not produce a forest for all prime powers $7<q\leq 67$, suggesting that the smaller examples are anomalies. The conjecture predicts the exact maximum induced-forest order for Paley graphs of square order beyond these small cases.