In the paper under review, the authors introduce the notion of residual normal crossing at a closed point of a variety \(X\) and observe that the \((p-1)\)-th power of a section \(s\) of \(K_X^{-1}\) (for a smooth variety \(X\) with the canonical bundle \(K_X)\) with a residual normal crossing at a closed point Frobenius splits the variety \(X\). This generalizes an earlier result due to Mehta and Ramanathan asserting that the \((p-1)\)-th power of a section \(s\) of \(K_X^{-1}\) with normal crossing divisor at some closed point Frobenius splits \(X\). In the present paper, it is further shown that for any parabolic subgroup \(Q\) of a simple group \(G\) of classical type or of type \(G_2\), \(K^{-1}_X\) (where \(X=G/Q)\) admits a section with residual normal crossing at the base point. This provides another proof of the Frobenius splitting of such an \(X\). Further the authors conjecture that for any homogeneous space \(X\) in characteristic \(p>0\), \(E\) is compatibly split in the blow-up \(B(X\times X)\) of \(X\times X\) along the diagonal \(\Delta\) with exceptional divisor \(E\). This is equivalent to showing that there exists a splitting of \(X\times X\) which vanishes to order \((\dim X)(p-1)\) generically along \(\Delta\). It is shown that this conjecture implies the Wahl conjecture on the surjectivity of the Gaussian map in characteristic \(p\). Recall that this was proved by the reviewer in characteristic 0 earlier [\textit{S. Kumar}, Am. J. Math. 114, No. 6, 1201-1220 (1992; Zbl 0790.14015)].