Modular transformation conjecture for the genus-two signature

Let $\sigma_2(q/p)$ denote the genus-two signature at a rational number $q/p$. For coprime odd integers $0<q<p$, consider the transformed fraction $q/(2q+p)$. Modular transformation conjecture. For any coprime odd integers $0<q<p$, one has

$$\sigma_2\Bigl(\frac{q}{2q+p}\Bigr)-\sigma_2\Bigl(\frac{q}{p}\Bigr)=2q^2+2pq+p^2-1.$$

This is an arithmetic analogue of the modular transformation law for the limiting function $\Lambda(\theta)$, suggested by the asymptotics of the genus-two signature along continued-fraction convergents. The identity was checked by computer for all $0<q<p<100$, but no proof or general resolution is given here.