Assumption-free discrepancy and vanishing conjecture for derived blow-ups

Let $f:X\to Y$ be a closed embedding of derived schemes, let $\mathbb{B}l_f$ be the derived blow-up, and let $pr_f:\mathbb{B}l_f\to Y$ be its projection. Denote the determinant line bundles of the cotangent complexes of $Y$ and $\mathbb{B}l_f$ by $K_Y$ and $K_{\mathbb{B}l_f}$, respectively. The discrepancy formula and Grauert–Riemenschneider vanishing theorem in question currently assume that $\mathbb{B}l_f$ is smooth and that $\pi_0(Y)-\pi_0(X)$ is nonempty.

Assumption-free discrepancy and vanishing conjecture. The assumptions that $\mathbb{B}l_f$ is smooth and that $\pi_0(Y)-\pi_0(X)$ is nonempty can be removed from the discrepancy formula and the corresponding Grauert–Riemenschneider vanishing theorem.

Thus the stated discrepancy formula and vanishing conclusion should remain valid without those two assumptions. The source presents this as a proposed removal of hypotheses; it does not provide a proof in the cited passage.