An independence of the MIN principle from the PHP principle

The minimization principle $\mathsf{\text{MIN}}(◃)$ studied in bounded arithmetic says that a strict linear ordering $◃$ on any finite interval $[0,\dots ,n)$ has the minimal element. We shall prove that bounded arithmetic theory ${\mathsf{\text{T}}}_2^1(◃)$ augmented by instances of the pigeonhole principle for all ${\mathrm{\Delta }}_1^b(◃)$ formulas does not prove $\mathsf{\text{MIN}}(◃)$.