The folklore conjecture on simultaneous rank-zero twists and Tate–Shafarevich orders

For positive integers $m$ and $k$, consider elliptic curves over the rationals and, for a prime $p$, their $p$-twists. Folklore conjecture. For any positive integers $m$ and $k$, there are pairwise non-isogenous elliptic curves $E^1,\dots,E^m$ defined over the rationals such that the Mordell–Weil group of the $p$-twist of each $E^i$ has rank zero for a positive proportion of primes $p$, and

$$|\text{Ш}(E^i)|=k^2,$$

for all $i=1,\dots,m$. This optimistic conjecture asks for arbitrarily large collections of pairwise non-isogenous curves with simultaneous positive-proportion rank-zero prime twists and prescribed square Tate–Shafarevich order; no resolution is given.