Generative models aim to approximate an unknown probability distribution μ on Rd using a finite sample of independent draws from μ. Motivated by variance-preserving score-based diffusion models, we introduce a new diffusion-based transport plan on path space that is optimal with respect to a criterion combining entropy minimization and stabilization of the quadratic variation. The resulting transport plan can be interpreted as an interpolation between the Schrödinger bridge and the Bass solution from martingale optimal transport. The proposed method has a computational complexity comparable to that of state-of-the-art approaches, while yielding a significant improvement in generation quality.