Given two SDEs $X^1$, $X^2$ :

$$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$

where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$. Can we prove $\mathbb P[\inf_{0\le s\le t}X^1_s>0]\ge \mathbb P[\inf_{0\le s\le t}X^2_s>0]$ for all $t\ge 0$?