The lower-bounds phase-transition conjecture

Suppose $T$ is a theory containing $\mathrm{I}\Sigma_1$, $l$ is nondecreasing, and $M_f$ is a computable function for every computable $f$, satisfying the three properties stated below:

1. $T \nvdash \forall x\,\exists y\,M_{\mathrm{id}}(x)=y$.
2. If $f(i)\leq g(i)$ for all $i\leq M_g(x)$, then $M_f(x)\leq M_g(x)$.
3. There exists $x$ such that $k\mapsto l(k)$ is eventually bounded by $k\mapsto M_k(x)$.

Lower-bounds phase-transition conjecture. Under these assumptions,

$$T \nvdash \forall x\,\exists y\,M_{l^{-1}}(x)=y.$$

The conjecture proposes a general mechanism for phase transitions in unprovability: when the upper-bound method fails because $l$ is a lower bound, the inverse threshold $l^{-1}$ should yield an unprovable totality statement. Its status is unclear from the supplied text.