Stable-completeness characterization of bases for exponential families

Let $1<p<\infty$, and let $U=\{u_k\}$ be an exponential-function family in $L^p(0,1)$, with the distance between normalized exponential-function families defined by the displayed quantity in the source. Call $U$ stably complete (respectively, stably incomplete) if there is a sufficiently small $\varepsilon>0$ such that completeness (respectively, incompleteness) is preserved for every exponential-function family $\widetilde U$ whose distance from $U$ is at most $\varepsilon$. Stable-completeness characterization conjecture. An exponential-function family is a basis in $L^p(0,1)$ if and only if it is uniformly minimal and stably complete; it is a basis in its span if and only if it is uniformly minimal and stably incomplete. When $p=2$, basicity should be replaced by unconditional basicity. The conjecture seeks a characterization of basis properties through uniform minimality and stability under perturbations. The source presents it as a conjectural connection and gives no evidence of resolution.