Quartic Legendre multiplier sequence region conjecture

Consider the sequence

$$\{(k^2+k)^2+b(k^2+k)+c\}_{k=0}^{\infty}.$$

A pair $(b,c)$ lies in the multiplier sequence region when this sequence is a multiplier sequence for the Legendre basis. Quartic Legendre multiplier sequence region conjecture. The sequence is a multiplier sequence for the Legendre basis if and only if $(b,c)$ lies in the region of the $bc$-plane bounded by the $b$-axis, the parabola

$$c=\frac{(b+2)^2}{8}-4,$$

and the curve

$$0=b^5-(c+15)b^4+4(c+12)b^3+8(c^2+24c+19)b^2-16(10c^2+101c+33)b-16(c^3-50c^2-185c+63).$$

The conjecture gives a proposed exact parameter region for quartic multiplier sequences in the Legendre basis; the source motivates it through an analysis of the symbol curve and numerical or graphical evidence, but provides no proof or resolution.