Only Forwards In Time

There is no price at all. Query and single-line update are $O(\log n)$, and advancing $t$ is $O(\log n)$ worst case, so the structure strictly dominates a Li Chao tree on every axis.

$t$ may only increase. Query and single-line update are $O(\log n)$, and advancing $t$ is $O(\log^2 n)$ amortized, an amortization that exists only because the certificates melt in one direction.

$t$ may move either way. Query and single-line update are $O(\log n)$, and advancing or rewinding $t$ is $O(n)$ per step, since the certificates must be recomputed whenever the clock runs backwards.

Lines must be inserted in order of increasing slope. Query and single-line update are $O(\log n)$, and advancing $t$ is $O(\log^2 n)$ amortized, an amortization that exists only because the slopes arrive in order.