Legendre-symbol triangulation polynomial congruence

Let $p$ be an odd prime, and let $E_p^+$ and $E_p^-$ be the convex $(p+1)$-near-edges associated respectively to the sequences of Legendre symbols and their negatives. Let $p_{E_p^+}$ and $p_{E_p^-}$ denote their maximal triangulation polynomials. For a polynomial $P(t)$ of the form

$$t p_{E_p^+}^2,\qquad t p_{E_p^-}^2,\qquad\text{or}\qquad t p_{E_p^+}p_{E_p^-},$$

and with $G_C(t)$ and the pairing $\langle\,\ \rangle_t$ defined as in the source's formulas, Legendre-symbol triangulation polynomial conjecture.

$$\langle P(t),t^2G_C(t)\rangle_t\equiv\left(\frac{-1}{p}\right)\pmod p =\begin{cases}1& p\equiv1\pmod4,\\-1& p\equiv3\pmod4. \end{cases}$$

This conjecture concerns congruences detected by maximal triangulation polynomials of near-edges constructed from Legendre symbols. The source reports computational evidence for all odd primes $p<200$, but provides no proof or later resolution in the supplied material.