Mechanism of Fritsch–Buttenberg–Wiechell rearrangement
As your heading correctly states, this reaction take the path of Fritsch–Buttenberg–Wiechell rearrangement (Wikipedia).   The reaction mechanism of Fritsch–Buttenberg–Wiechell rearrangement (FBW) is described as:


  The strong base deprotonates the vinylic hydrogen, which after alpha-elimination forms a vinyl carbene. A 1,2-aryl migration forms the 1,2-diaryl-alkyne product. The mechanism of the FBW rearrangement was a subject of on-surface studies where the vinyl radical was visualized with sub-atomic resolution (Ref.1).


However, if you carefully read this reference, you'd realize they use special method to generate diradical from gem-dibromo-alkene stepwise. Thus, the proposed mechanism for every FBW-type reactions. Earlier, A. A. Bothner-By (Ref.2) from  Harvard reported that in a diaryl-substituted system of a bromoalkene (e.g., see structures 1 and 4 in the scheme 1), the major migrating group is the aryl moiety located anti to the halogen (the fact, which has been proven by $\ce{^{14}C}$-labeling experiments). Particularly, this migration of an aryl group between multiply-bonded carbons is one of the several unusual features of the rearrangement (Ref.2): 

 

Yet, there are plethora of data to support the 1,2-phenyl group migration through single bonds (e.g., the benzilic acid rearrangement), and at least one example for migration of an aryl group between multiply-bonded carbons (e.g., the Beckmann rearrangement). Thus, based on these facts, I suggests following arrow-pushing mechanism, which would be very reasonable:



References:


N. Pavliček, P. Gawel, D. R. Kohn, Z. Majzik, Y. Xiong, G. Meyer, H. L. Anderson, L. Gross, “Polyyne formation via skeletal rearrangement induced by atomic manipulation,” Nature Chemistry 2018, 10, 853–858 (https://www.nature.com/articles/s41557-018-0067-y).
Aksel A. Bothner-By, “Stereochemistry of the rearrangement involving migration between multiply-bonded carbons,” J. Am. Chem. Soc. 1955, 77(12), 3293–3296 (https://doi.org/10.1021/ja01617a042).