$\mathrm{Almost}\text{-}\oplus\mathrm{P} = \mathrm{BP}\cdot\oplus\mathrm{P}$ and a Random-Oracle Proof of Toda's Theorem

Using the recent exponential correlation bounds of Chattopadhyay, Hatami, Lee, Lovett, Tal and Viola between $\mathbb{F}_2$-polynomials and the XOR of majorities, we show that $\mathrm{Almost}\text{-}\oplus\mathrm{P} = \mathrm{BP}\cdot\oplus\mathrm{P}$, where $\mathrm{Almost}\text{-}\oplus\mathrm{P}