The interval conjecture for second-level universal lower bounds

For each dimension $n$, let $D(n,\tau)$ and $D(n,\tau+1)$ be the endpoint cardinalities associated with consecutive levels, and consider those $N$ in the interval

$$N\in [D(n,\tau),D(n,\tau+1)]$$

for which second-level bounds exist.

Interval conjecture. For every dimension $n$, the values $N\in [D(n,\tau),D(n,\tau+1)]$ for which second-level bounds exist form an interval.

The conjecture concerns the observed organization of second-level universal lower bounds across cardinalities; the source formulates it as a concluding conjecture and provides examples, but no resolution.