Recent Progress on Conjecture $\mathcal{O}$ and Its Variants: A Survey^†

Comments 1: [line 2 of abstract. Why to introduce the shorten version GGI if you never use it?]

Response 1: Thank you for pointing this out. We agree with this comment. Therefore, we have removed 'GGI' from the line 2 of the abstract.[we removed ‘GGI’ from the line 2 of the abstract, page 1, paragraph 1, line 2.]

Comments 2: [lines 31, 32, 33. So badly written... please split them into two different sentences.]

Response 2: Thank you for pointing this out. We agree with this comment. We corrected the misspelled word 'opeator' in the text and, as per your suggestion, split the sentence into two parts. 

Comments 3: [overall, I feel that main references are missing, and more context could be given to conjecture O. Could you improve this?]

Response 3: Agree. Thank you for pointing this out, We have added contextual background following Conjecture O. First, we included Rietsch' s findings on the properties of the quantum product operator in the quantum cohomology of Grassmannians. Second, we have included an explanation of the meaning of "O" in Conjecture O. Third, we incorporated the result by Galkin, Golyshev, and Iritani proving that for toric Fano manifolds, Gamma Conjecture I holds whenever Conjecture(property) O is satisfied.

Comments 4: [The line after the statement of Property 2.1 is badly typed.]

Response 4: Thank you for pointing this out. We have retyped this paragraph.

Comments 5: [I want more references in pages 2 and 3. For each statement a reference.]

Response 5: Thank you for pointing this out. We have added citations [9,14,19,9] in page 2, citations [20,7,21] in page 3.

Comments 6: [Section 3: I find somehow bad that no reference to the work of A. Givental is given. He was the first to prove the toric mirror theorem.]

Response 6: You are completely right, and we appreciate this insightful comment. We have added citations to Givental's results in line 153.