Rakočević's conjecture on continuity of the Moore–Penrose inverse

Let $A$ be a unital Banach algebra, and let $A_{\mathrm{MP}}$ denote the set of Moore–Penrose invertible elements of $A$. For $a\in A_{\mathrm{MP}}$, write $a^\dag$ for its Moore–Penrose inverse. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence in $A_{\mathrm{MP}}$ converging to $a\in A_{\mathrm{MP}}$. Rakočević's conjecture. Then

$$a_n^\dag\to a^\dag \quad\text{if and only if}\quad \sup_{n\in\mathbb{N}}\lVert a_n^\dag\rVert<\infty.$$

This conjecture concerns norm continuity of the Moore–Penrose inverse in unital Banach algebras. The source attributes it to Rakočević and presents the stated result as being established for certain Banach algebras, including all $L^p$-operator algebras; its status as a conjecture in the full generality stated here is not resolved by the supplied context.