Florence's lifting conjecture for Borel representations

Let $p$ be a prime number, let $F$ be a field containing a primitive $p^2$-th root of unity, and let $n\geq 1$ be an integer. Let $\Gamma_F$ denote the absolute Galois group of $F$, and let $B_n$ be the Borel subgroup of upper-triangular matrices in $\operatorname{GL}_n$. Florence's conjecture. Every continuous homomorphism

$$\Gamma_F\to B_n(\mathbb F_p)$$

lifts to a continuous homomorphism

$$\Gamma_F\to B_n(\mathbb Z/p^2\mathbb Z).$$

This is a Borel-valued refinement of the question of whether mod-$p$ Galois representations lift modulo $p^2$. The conjecture is stated for fields containing a primitive $p^2$-th root of unity; the surrounding discussion records affirmative results in several special cases, while the general assertion remains open.