Siegel eigenform congruence from a symmetric-square $L$-value

Let $N\geq 1$ be square-free, let $\chi$ be a quadratic character mod $N$, let $j\geq 0$ and $k\geq 3$. Suppose $f\in S_{j+k}(\Gamma_0^{(1)}(N),\chi)$ is an eigenform with $\operatorname{ord}_{\mathfrak{p}}(L_{\operatorname{alg}}(\operatorname{Sym}^2(f),j+2k-2))>0$ for some prime $\mathfrak{q}$ of $\mathbb{Q}_f$ lying above a rational prime $q>2(j+k)-1$. Symmetric-square congruence conjecture. Then there exists an eigenform $F\in S_{j,k}(\Gamma_0^{(2)}(N),\chi)$ and a prime $\mathfrak{q}'\mid\mathfrak{q}$ of $\mathbb{Q}_{f,F}$ such that

$$b_{1,p^2}\equiv a_p^2-\chi(p)p^{j+k-1}-p^{j+2k-5}+p^{j+2k-3}+p^{j+1}\pmod{\mathfrak{q}'},$$

for all $p\nmid N$, where $T_p(f)=a_p f$ and $T_{1,p^2}(F)=b_{1,p^2}F$. The conjecture generalizes the congruences observed in the paper between elliptic modular forms and Siegel modular forms; the stated special case is reported as proved in the example immediately preceding it, while the general assertion is presented as conjectural.