The isomorphism conjecture for true unprovable sentences

Let $\mathcal{S}$ and $\mathcal{T}$ be consistent theories, with $\mathcal{T}$ strictly stronger than $\mathcal{S}$, and let $\mathfrak{I}$ be a collection of sentences unprovable in $\mathcal{T}$. For $\phi\in\mathfrak{I}$, write $\mathcal{T}\nvdash\phi$ when $\mathcal{T}$ does not prove $\phi$, and interpret the displayed bounds as bounds on proof lengths. Isomorphism conjecture. The conjecture holds when $\mathfrak{I}$ consists of any true sentences $\phi$ unprovable in $\mathcal{T}$: if $\mathcal{S}$ cannot prove that $\mathcal{T}$ cannot prove $\phi$, then $\mathcal{S}$ cannot do so with a proof of length $t^{\mathcal{O}(1)}$ when the asserted absence concerns proofs of length at most $t$. The conjecture is false if $\mathfrak{I}$ includes all unprovable sentences, because false unprovable sentences can provide counterexamples; its restriction to true sentences is proposed as an open possibility.