Summary: We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting \(J_{k-1}(s)=\int^s_0 t^{k-1} e^{-\frac{t^2}{2}}dt\) and \(c_{k-1}=J_{k-1}(+\infty )\), we conjecture that the function \(F:[0,1]\rightarrow \mathbb{R} \), given by \[F(a)= \sum_{k=1}^n 1_{a\in E_k}\cdot (\beta_k J_{k-1}^{-1}(c_{k-1} a)+\alpha_k) \] (with an appropriate choice of a decomposition \([0,1]=\bigcup_i E_i\) and coefficients \(\alpha_i, \beta_i)\) satisfies, for all symmetric convex sets \(K\) and \(L\), and any \(\lambda \in [0,1]\), \[ F\left (\gamma (\lambda K+(1-\lambda )L)\right )\geq \lambda F\left (\gamma (K)\right )+(1-\lambda ) F\left (\gamma (L)\right ).\] We explain that this conjecture is “the most optimistic possible”, and is equivalent to the fact that for any symmetric convex set \(K\), its Gaussian concavity power \(p_s(K,\gamma )\) is greater than or equal to \(p_s(RB^k_2\times \mathbb{R}^{n-k},\gamma )\), for some \(k\in \{1,\dots ,n\} \). We call the sets \(RB^k_2\times \mathbb{R}^{n-k}\) round \(k\)-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see S. Heilman [Am. J. Math. 143, No. 1, 53–94 (2021; Zbl 1462.53003)]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set \(K\) in \(\mathbb{R}^n\), \[ p_s(K,\gamma )\geq \sup_{F\in L^2(K,\gamma )\cap Lip(K):\,\int F=1} \left (2T_{\gamma }^F(K)-Var(F)\right )+\frac{1}{n-\mathbb{E}X^2},\] where \(T_{\gamma }^F(K)\) is the \(F-\) torsional rigidity of \(K\) with respect to the Gaussian measure. Moreover, the equality holds if and only if \(K=RB^k_2\times \mathbb{R}^{n-k}\) for some \(R>0\) and \(k=1,\dots ,n\). As a consequence, we get \[ p_s(K,\gamma )\geq Q(\mathbb{E}|X|^2, \mathbb{E}\|X\|_K^4, \mathbb{E}\|X\|^2_K, r(K)), \] where \(Q\) is a certain rational function of degree \(2\), the expectation is taken with respect to the restriction of the Gaussian measure onto \(K, \|\cdot \|_K\) is the Minkowski functional of \(K\), and \(r(K)\) is the in-radius of \(K\). The result follows via a combination of some novel estimates, the \(L2\) method (previously studied by several authors, notably A. V. Kolesnikov and E. Milman [ J. Geom. Anal. 27, No. 2, 1680–1702 (2017; Zbl 1372.53040); Am. J. Math. 140, No. 5, 1147–1185 (2018; Zbl 1408.53047); Lect. Notes Math. 2169, 221–234 (2017; Zbl 1366.60056); Local \(L^p\)-Brunn-Minkowski inequalities for \(p < 1\). Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1502.52002)], A. V. Kolesnikov and the author [Adv. Math. 384, Article ID 107689, 23 p. (2021; Zbl 1479.52014)], Hosle, Kolesnikov and the author [J. Hosle et al., J. Geom. Anal. 31, No. 6, 5799–5836 (2021; Zbl 1469.52008)], A. Colesanti [Commun. Contemp. Math. 10, No. 5, 765–772 (2008; Zbl 1157.52002)], Colesanti, the author and Marsiglietti [A. Colesanti et al., J. Funct. Anal. 273, No. 3, 1120–1139 (2017; Zbl 1369.52013)], A. Eskenazis and G. Moschidis [J. Funct. Anal. 280, No. 6, Article ID 108914, 20 p. (2021; Zbl 1456.52011)]) and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the “convex set version” of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function \(F\) whose composition with \(\gamma (K)\) is concave in the Minkowski sense for all symmetric convex sets.