Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$ According to Theorem 1.7 of Mark-Tosun's paper, the Brieskorn sphere $\Sigma(2,3,6m+1)$ admits two tight contact structure $\xi_{i}\ (i=0,1)$. They are both Stein fillable and they are contactomorphic (but not isotopic).
My question:

Consider the Ozsváth-Szabó contact invariant $c(\xi_{i})\in HF^{+}(-\Sigma(2,3,6m+1))$. We know it is non-zero by Stein fillability, but do we know whether its image under the natural map 
  $$
HF^{+}(-\Sigma(2,3,6m+1))\rightarrow HF^{\text{red}}(-\Sigma(2,3,6m+1))
$$
  is non-zero?

Such result can be shown, e.g., if $\xi_{i}$ bounds a non negative definite Stein domain.
 A: (Note: I might have screwed up orientations, so take everything with a grain of salt.) I will write an argument for the case $m=1$, showing that the reduced contact invariant $c^{\rm red}(\xi)$ is non-zero. I think that the argument(s) can be tweaked to work for $m>1$, as well, but I won't try to write it down properly.
Let me suppress some things from the notation: I'll call $\Sigma = \Sigma(2,3,7)$, and let $\xi$ denote either of $\xi_0$ or $\xi_1$. (The statement is invariant under self-diffeomorphisms of $\Sigma$.)
Mark and Tosun give an explicit surgery presentation of $\xi$ as Legendrian surgery along a right-handed trefoil with Thurston-Bennequin $0$. This gives a Stein cobordism $W: S^3 \leadsto \Sigma$, and the cobordism map $F: = F^+_{\overline W}: HF^+(-\Sigma) \to HF^+(-S^3)$ maps $c(\xi)$ to $c(\xi_{\rm std})$.
The key point is showing that $F$ vanishes on the tower in $HF^+(\Sigma)$ (i.e. it vanishes for sufficiently large degrees). This can be seen in two ways.
The first is to show that spin$^c$ structures on $W$ come in conjugate pairs, and all their contributions cancel in pairs on the tower. (Note that they can -and do, a posteriori- move elements outside the tower.)
The second is that $F$ fits into an exact triangle:
$$ \dots \to HF^+(-\Sigma_0) \to HF^+(-\Sigma) \to HF^+(-S^3)\to \dots$$
where $\Sigma_0$ is the 0-surgery along the (right-handed) trefoil.
It's easy to see (basically by rank count) that the map $HF^+(-\Sigma_0) \to HF^+(-\Sigma)$ must be surjective onto the tower, which must then be killed by $F$.
Either way, if $c^{\rm red}(\xi)=0$, then $F(c^{\rm red}(\xi)) = 0 \neq c(\xi_{\rm std}))$.
A: Brieskorn-Pham calculated the signature of the smoothing of the Milnor fiber of that Brieskorn singularity (in this case the smoothing of the complex singularity $x^2+y^3+z^{6m+1}=0$ or the symplectic branched double cover over the smoothing of the algebraic surface bounding $T(3,7)$ ). In the case of $M(2,3,6m+1)$, this is non-definite. See remark 4.6 of http://www.maths.ed.ac.uk/~aar/papers/nemethi1.pdf for Brieskorn's formula.
In the case of $\Sigma(2,3,6m+1)$, the contact structure has homotopy type $\theta=-2$. As $c_1=0$ for $M(2,3,6m+1)$ (this can be seen by the double cover representation), Gompf's formula for $\theta$, $c_1^2-3\sigma(X)-2\chi(X)=\theta$ gives 
$$\sigma(M(2,3,6m+1))=-8m$$
