Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} Var(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$
\mathbb{P}\left[ \sum_{i=1}^{n}X_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2 + \frac{2}{3}xy}\right)}.
$$

A similar bound (albeit for independent random variables) is given in Corollary 2 in https://epubs.siam.org/doi/abs/10.1137/1134032. I have seen that exponential inequalities for sums of independent random variables can be extended to martingales generally. 

Question 2: If the bound given in question 1 doesn't hold, does any other similar exponential inequality exist for the LHS?
I have came across Freedman's inequality (Theorem 1.6 in [Freedman (1975)][1]) which deals with similar quantities but it contains $Var(X_i | \mathcal{F}_{i-1})$. As seen from the above, I would rather have the bound in terms of $Var(X_i)$. 

Thank you for your time and consideration. 

[1]: https://projecteuclid.org/euclid.aop/1176996452