Inconsistency in Witten's superstrings [closed]

Superstrings volume 1 green, Schwarz witten

After doing some calculation, in wittens convention for the energy mkmentum tensor of the RNS action I somehow get his version of the energy momentum tensor is not consistent with his later stated convention in LC coordinates

Witten uses the mostly plus metric with the stated gamma matricies as: $\rho^0=\begin{pmatrix} 0&-i\\i&0\end{pmatrix}$ $\rho^1 =\begin{pmatrix} 0&i\\i&0\end{pmatrix}$ In LC coordinates with wittens formula $T_{ab}= \partial_a x^{\mu} \partial_b x_{\mu} +\frac{i}{4} \bar{\psi}^{\mu}\rho_{a}\partial_b \psi_{\mu}+\frac{i}{4} \bar{\psi}^{\mu}\rho_{b}\partial_a\psi_{\mu}$

$T_{++}= \partial_+ x^{\mu} \partial_+ x_{\mu} +\frac{i}{4} \bar{\psi}^{\mu}\rho_{+}\partial_+\psi_{\mu}+\frac{i}{4} \bar{\psi}^{\mu}\rho_{+}\partial_+\psi_{\mu}$ $T_{++}= \partial_+ x^{\mu} \partial_+ x_{\mu} +\frac{i}{2} \bar{\psi}^{\mu}\rho_{+}\partial_+\psi_{\mu}$

I have that following the jacobian in LC coordinates:

$\rho_+ = J^\alpha_+ \rho_\alpha$ $\rho_+ = J^0_+\rho_0 +j^1_+ \rho_1$ $\rho_+ = -J^0_+\rho^0 +j^1_+ \rho^1$ $\rho_+ =-\frac{1}{2}\rho^0 +\frac{1}{2} \rho^1$

$\rho_+=\begin{pmatrix} 0&i\\0&0\end{pmatrix}$

Where $\rho^0\rho_+ = \begin{pmatrix} 0&-i\\i&0\end{pmatrix}\begin{pmatrix} 0&i\\0&0\end{pmatrix} =\begin{pmatrix} 0&0\\0&-1\end{pmatrix}$ We have $T_{++}= \partial_+ x^{\mu} \partial_+ x_{\mu} +\frac{i}{2} \\{\psi^T}^{\mu}\rho^0\rho_{+}\partial_+\psi_{\mu}$

The second term gives $+\frac{i}{2}\begin{pmatrix} \psi^\mu_-&\psi^\mu_+\end{pmatrix}\begin{pmatrix} 0&0\\0&-1\end{pmatrix}\begin{pmatrix}\partial_+ {\psi_\mu}_-\\ \partial_+ {\psi_\mu}_+\end{pmatrix}$

$-\frac{i}{2}\begin{pmatrix} \psi^\mu_-&\psi^\mu_+\end{pmatrix}\begin{pmatrix}0\\ \partial_+ {\psi_\mu}_+\end{pmatrix}$ $-\frac{i}{2}\psi^\mu_+\partial_+ {\psi_\mu}_+$ Giving us $T_{++}=\partial_+x^\mu\partial_+x_\mu-\frac{i}{2}\psi^\mu_+\partial_+ {\psi_\mu}_+$

Which is different from his stated formula: $T_{++}=\partial_+x^\mu\partial_+x_\mu+\frac{i}{2}\psi^\mu_+\partial_+ {\psi_\mu}_+$

So now I am asking. Is this calculation correct? If not where is the mistake, if yes why is there a discrepancy between wittens conventions?