General comments.

We are grateful to the referee for their careful reading and thoughtful comments.

Regarding the comment about implementing the bootstrap procedure in the Virasoro minimal string (VMS): precisely because the string amplitudes in VMS are simply polynomials, they have no interesting analytic structure that we can leverage in implementing our analytic bootstrap procedure. It would be interesting to actually prove this polynomiality of the volumes directly from the worldsheet, without having to appeal to the 3d gravity presentation.

We certainly agree that a better understanding of the relation between minimal Liouville gravity, the VMS, and the $\mathbb{C}$LS, perhaps stemming from the limits that relate their underlying worldsheet CFTs, is an interesting future line of research. At present, we do not know if such limits uplift to the full string theory and their string amplitudes.

Regarding the comment about footnote 13, we have narrowed the scope of the footnote to emphasize the fact that correlation functions of degenerate representations in timelike Liouville CFT are not obtained from those of generic representations by analytic continuation, without foreclosing the possibility of a ground ring in the Virasoro minimal string.

Suggestions for improvement.

1. We referenced appendix B in the paragraph right before section 6.2.

2. We appreciate the referee's careful reading of our manuscript and their suggestions regarding the use of footnotes. While we understand the concern, we believe that the footnotes serve to keep the main text focused while still providing relevant context and clarifications.
   
   That said, we have carefully reviewed our use of footnotes and have made adjustments where appropriate to improve readability. In particular we have removed footnote 2 and included the quantum volumes in the main text.

3. Since in the matrix integral paper we present the quantum volumes alongside the combination of $\sin$'s associated to the different vertices we would rather include the quantum volumes in the sum. We do not feel that it detracts from the readability of the equations.

4. Addressed.

5. We thank the referee for their comment. While the table of contents provides an overview, we believe that a brief outline in the introduction helps guide the reader by highlighting the key ideas and logical flow of the paper.

6. Thank you for the comment. We have modified the statement appropriately.

7. We have added a clarifying sentence.

8. We have removed the word convention, but this is meant to refer to a choice of normalization for the Liouville vertex operators $V_p$, together with the integration measure in the OPE between two such operators, expressed as an integral over intermediate states.

9. Done.

10. The ghost contributions have been taken into account in those equations. In the case of the torus one-point amplitude, it is the ghost correlator that gives rise to the Dedekind eta factor.

11. We clarified the sentence.

12. Thank you for pointing out this typo. We have reincorporated the word "second."

13. Using a tilde for right-moving quantities is not really unconventional. For example, Polchinski's volumes on String Theory follow this notation. We deliberately adopted this notation in order to avoid the possible confusion that a bar for right-moving quantities does not imply a complex conjugation. We have added a footnote on page 10 explaining this convention. To make the point, consider the conformal block decomposition in Liouville CFT at complex central charge. While the right-moving conformal block involves the cross-ratio $\bar{z}$, it does not involve complex-conjugated external weights, or momenta $p_j^*$.

14. We added the word nontrivial.

15. Yes. Fixed.

16. We believe the statement is readable as is.

17. We agree. We would love to be able to prove this property directly from the worldsheet but have not succeeded in doing so.