Exceptional-zero conjecture for the symmetric-square p-adic L-function

Let $\mathbf{f}$ be the Hilbert modular form in the paper, let $F$ be its totally real coefficient field, and let $m$ be the parameter for which the trivial zero occurs at $s=m+1$. Define

$$L_p(s,\operatorname{Sym}^2(\mathbf{f}))=G(u^s-1,\operatorname{Sym}^2(\mathbf{f})),$$

where $u$ generates $1+p\mathbb{Z}_p$, and let $E^*(m+1,\mathbf{f})$ be the product of the nonzero factors of $E(m+1,\mathbf{f})$. If $g$ is the number of primes above $p$ in $F$, the symmetric-square exceptional-zero conjecture. Then

$$\operatorname{ord}_{s=m+1}L_p(s,\operatorname{Sym}^2(\mathbf{f}))=g$$

and

$$\left.\frac{\mathrm{d}^gL_p(s,\operatorname{Sym}^2(\mathbf{f}))}{\mathrm{d}s^g}\right|_{s=m+1}=\mathcal{L}(\operatorname{Sym}^2(\mathbf{f}))C E^*(m+1,\mathbf{f})\frac{L(m+1,\operatorname{Sym}^2(\mathbf{f}))}{\Omega(\mathbf{f},m+1)}.$$

Here $\mathcal{L}(\operatorname{Sym}^2(\mathbf{f}))$ is a nonzero term determined by the Galois cohomology of $\operatorname{Ind}_F^{\mathbb{Q}}(\operatorname{Sym}^2\rho_{\mathbf{f}})$. This is the paper's explicit prediction for the order and leading term of the trivial zero when the Nebentypus is trivial.