Average character degree refinement of Gluck's conjecture

Let $G$ be a finite group, let ${\mathbf F}(G)$ denote its Fitting subgroup, and let $\operatorname{Irr}(N)$ be the set of irreducible characters of a normal subgroup $N$. For $\lambda\in\operatorname{Irr}({\mathbf F}(G))$, define

$$\operatorname{acd}(G\mid\lambda)=\frac{\sum_{\chi\in\operatorname{Irr}(G\mid\lambda)}\chi(1)}{|\operatorname{Irr}(G\mid\lambda)|},$$

where $\operatorname{Irr}(G\mid\lambda)$ is the set of irreducible characters of $G$ lying over $\lambda$. Average character degree refinement of Gluck's conjecture. If $G$ is solvable, then there exists a linear character $\lambda\in\operatorname{Irr}({\mathbf F}(G))$ such that

$$|G:{\mathbf F}(G)|\leq\operatorname{acd}(G\mid\lambda)^2.$$

This would strengthen Gluck's conjecture because $\operatorname{acd}(G\mid\lambda)\leq b(G)$. The source presents it as a proposed refinement, and its proof only establishes the weaker exponent bound $\alpha<2.596$; the conjecture remains open.