The Homological Time Complexity Relation

Let $L$ be a computational problem, let $T_L(n)$ denote its time complexity, and let $h(L)$ denote its homological complexity. Homological Time Complexity Relation. There exists a polynomial $p$ such that, for every computational problem $L$,

$$T_L(n)=\Omega\left(2^{h(L)\cdot\log n}\right).$$

Thus homological complexity is conjectured to provide an exponential lower bound on time complexity. The source gives no resolution or supporting result sufficient to establish this relation, so its status remains open.