The determinant identity for the half-space and full-space TASEP kernels

Let $K_1$ be the product of the first three matrices in the displayed factorization of the TASEP kernel, and let $K_2$ be the operator defined by

$$K_2:=\begin{pmatrix} {\texttt{b}_*}1_0{\texttt{b}_*}^{-1} & {\texttt{b}_*}1_0{\texttt{b}}^{-1}\partial {\texttt{b}_*}^{-1}1_0{\texttt{b}}\\ 0 & {\texttt{b}}^{-1}1_0{\texttt{b}} \end{pmatrix}.$$

Here the operators act on the indicated $L^2$ spaces. The determinant identity conjecture. Due to the special structure of ${\texttt{b}_*}$ and ${\texttt{b}}$, one should have

$$\det\left(I+\tfrac{1}{2}K_1\right)_{{L^2([0,\infty))}\times {L^2([0,\infty))}}=\det\left(I+K_1K_2\right)_{L^2(\mathbb{R})\times L^2(\mathbb{R})}.$$

This identity would relate the half-space kernel to the full-space TASEP formula. The preceding transformation is only formal because some intermediate operators are not well defined, although the resulting kernel makes sense; the conjectured determinant equality is not resolved in the supplied text.