Uniqueness of distribution conjecture for independent copies spanning an Orlicz sequence space

Let $1\le p<2$ and let $M$ be a $p$-convex and $2$-concave Orlicz function. A mean zero function $f$ has independent copies equivalent in $L_p$ to the unit vector basis of $l_M$ when those copies are equivalent to that basis in $L_p$; uniqueness up to equivalence near $0$ means that there is only one such function in this sense.

Uniqueness conjecture. If there is a unique (up to equivalence near $0$) mean zero function $f$ whose independent copies are equivalent in $L_p$ to the unit vector basis in $l_M$, then $M$ is $(p+\varepsilon)$-convex and $(2-\varepsilon)$-concave for some $\varepsilon>0$.

The conjecture asserts that the convexity and concavity conditions sufficient for uniqueness in the preceding theorem are also necessary. The surrounding discussion notes that the result is sharp in general, since uniqueness can fail for an Orlicz function equivalent to $t/\log(e/t)$ near zero.