Block-structure conjecture for norm-one projections with unconditional bases

Let $X$ be a strictly monotone Banach space with a $1$-unconditional basis, sufficiently different from a Hilbert space. Block-structure conjecture. Every norm-one projection on $X$ is a weighted conditional expectation operator, and every $1$-complemented subspace of $X$ is spanned by mutually disjoint elements. The preceding block-basis theorem gives this form for a class of explicitly constructed $1$-complemented subspaces, and the source postulates that it is the correct general description. The conjecture is presented as an open extension of the conditional-expectation picture to Banach spaces with bases.