A note on the independence of the injective pigeonhole principle from the uniform counting principle

In this technical note, we show that the relativized bounded arithmetic $T_2^1(R)$ equipped with $UCP({\mathrm{\Delta }}_1^b(R))$ cannot prove $injPH{P}_n^{n+1}(R)$. The result partially supports [K, 2024] Conjecture 1.