  Assuming the existence of certain large cardinal numbers, we prove that for
every projective filter $\mathscr F$ over the set of natural numbers,
$\mathscr{F}$-bases in Banach spaces have continuous coordinate functionals. In
particular, this applies to the filter of statistical convergence, thereby we
solve a problem by V. Kadets (at least under the presence of certain large
cardinals). In this setting, we recover also a result of Kochanek who proved
continuity of coordinate functionals for countably generated filters (Studia
Math., 2012).
