Equality of automorphic L-invariants across cohomology degrees

Let $F$ be a number field, let $b\mathfrak{p}d$ be a fixed prime above $p$, let $\pi$ be the automorphic representation and let $i$ and $s$ be the cohomological indices occurring in the construction. For every continuous homomorphism

$$\ell\colon F_{\mathfrak{p}}^{\ast}\to \mathbb{Q}_p,$$

and every sign character $\epsilon$, write $\mathcal{L}_{\ell}^{(i)}(\pi,{\mathfrak{p}})^{\epsilon}$ for the automorphic $\mathcal{L}$-invariant and $c^{(i)}_{\ell}(\pi)^{\epsilon}$ for the corresponding modular-symbol class, with $c^{(\ast)}$ denoting the class in the relevant cohomology degree. Automorphic $\mathcal{L}$-invariant conjecture. The following assertions are conjectured:

$$\mathcal{L}_{\ell}^{(i)}(\pi,{\mathfrak{p}})^{\epsilon}=\left(\mathcal{L}_{\ell}^{(0)}(\pi,{\mathfrak{p}})^{\epsilon}\right)^{\binom{s}{i}}.$$

$$c^{(\ast)}_{\ell}(\pi)^{\epsilon}=\mathcal{L}_{\ell}^{(0)}(\pi,{\mathfrak{p}})^{\epsilon}\cdot c^{(\ast)}_{\operatorname{ord}_{{\mathfrak{p}}}}(\pi)^{\epsilon}.$$

3. The $i$-th $\mathcal{L}$-invariant $\mathcal{L}_{\ell}^{(i)}(\pi,{\mathfrak{p}})^{\epsilon}$ does not depend on the character $\epsilon$.
4. More generally,

$$c^{(\ast)}_{\ell}(\pi)=\mathcal{L}_{\ell}^{(0)}(\pi,{\mathfrak{p}})^{\epsilon}\cdot c^{(\ast)}_{\operatorname{ord}_{{\mathfrak{p}}}}(\pi)$$

for one, and therefore every, sign character $\epsilon$.