In Davis, William J., and William B. Johnson. "Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces." Israel Journal of Mathematics 14.4 (1973): 353-367., the authors discuss the following problem

"If $M$ is a norming subspace of $X^*$ and $Y$ is an $M$-closed subspace of $X$, then is $M \cap Y^\perp$ a norming subspace of $(X/Y)^*$ (where $Y^\perp$ is identified with $(X/Y)^*$ in the canonical way)?"

In the paper, the authors conclude that if $X$ is not quasi-reflexive, then there exists a norming subspace $M$ such that the conclusion fails.

What I am interested in is whether the result can be salvaged in some way by adding some reasonable conditions on $M$, and I would like to know if there is anything in the literature that has addressed this. The papers citing this paper do not discuss this particular question any further.

Thank you for your time.