Does a traceless energy-momentum tensor really imply a massless field?

I read a paper where there was written that "a Traceless Energy-momentum Tensor implies a massless field", so I did a bit of calculations but I seems not really true, is it true?

So now I’m considering 3 Lagrangian density:

• GR, with $T^{mu nu}=-frac{1}{2}sqrt{-g}frac{delta mathcal{L}}{delta g_{mu nu}}$

1. Spin $1$ field (EM):

$$mathcal{L}_1=-frac{1}{4}F^{rho sigma}F_{rho sigma}sqrt{-g}$$

$$Rightarrow T_1^{mu nu}=F^{mu sigma}F_{rhosigma}g^{rhonu}-frac{1}{4}g^{mu nu}F^{rho sigma}F_{rho sigma}$$

2. Spin $frac{1}{2}$ field (Massless Dirac):

$$mathcal{L}_2=ibar{psi}left(gamma^{rho}partial_{rho}right)psisqrt{-g}$$

$$Rightarrow T_2^{mu nu}=ig^{mu nu}bar{psi}left(gamma^{rho}partial_{rho}right)psi-ibar{psi}left(gamma^{mu}partial^{nu}+gamma^{nu}partial^{mu}right)psi$$

3. Spin $0$ field (Massless Real Klein-Gordon):

$$mathcal{L}_3=frac{1}{2}partial_{rho}phi partial^{rho}phisqrt{-g}$$

$$Rightarrow T_3^{mu nu}=frac{1}{2}g^{mu nu}partial_{rho}phi partial^{rho}phi – partial^{mu}phipartial^{nu}phi$$

In 4 Dimension the Trace $T=g_{mu nu}T^{mu nu}$ is :
$$T_1=0$$
$$T_2=2ibar{psi}left(gamma^{rho}partial_{rho}right)psineq0$$
$$T_3=partial_{rho}phi partial^{rho}phineq0$$

• Classical Field Theory, with $T^{mu nu}=frac{delta mathcal{L}}{partialleft(partial_mu theta right)}partial^nu theta-mathcal{L}g^{mu nu}$

1. Spin $1$ field (EM) ($theta=A^{rho}$):

$$mathcal{L}_1=-frac{1}{4}F^{rho sigma}F_{rho sigma}$$

$$Rightarrow tilde{T}_1^{mu nu}=F^{mu sigma}F_{rhosigma}g^{rhonu}+frac{1}{4}g^{mu nu}F^{rho sigma}F_{rho sigma}$$

But then in this case the energy-momentum tensor is no more traceless, in fact:
$$tilde{T}_1=2 F^{rho sigma}F_{rho sigma}$$
(For completeness this is the Simmetrized Belinfante Form, I could have not simmetrised it but I would have obtained $tilde{T}_1=frac{3}{2} F^{rho sigma}F_{rho sigma}neq 0$ anyway).

• Final Questions:

So what’s the meaning of the Trace $tilde{T}_1$ not being zero in the Classical Field Theory case?

Why $tilde{T}_1^{mu nu}neq T_1^{mu nu}$? Shouldn’t they be the same? Do they describe different physics?

So "a Traceless Energy-momentum Tensor implies a massless field" is it true only for the EM field in GR?