Neither SUMIFS nor SUMPRODUCT deliver the required result

The RANGE CRITERIA are in column B, C, D and E, rows 4-43.
The CRITERIA also in column B, C, D and E in row 52,
The SUM RANGE in column F, rows 4-43.
=SUMIFS(F$4:F$43,$B$4:$B$43,$B52,
$C$4:$C$43,$C52,
$D$4:$D$43,$D52,
$E$4:$E$43,$E52)

The SUMIFS works like a charm when all CRITERIA (B52, C52, D52, E52) are correctly populated. If however (i) one or more of the CRITERIA is not populated or (ii) is populated with a CRITERIA that doesn’t match with any of the fields in the respective RANGE CRITERIA, the result is zero.
What I would like the formula to do is SUM, based on one, two, three or all four criteria.

To no avail, I’ve also tried to get the required result a SUMPRODUCT formula:
=SUMPRODUCT(($B$4:$B$43=$B52)*
($C$4:$C$43=$C52)*
($D$4:$D$43=$D52)*
($E$4:$E$43=$E52)
*F$4:F$43)
The SUMPRODUCT results are identical as with SUMIFS: works when all CRITERIA are correctly populated, don’t deliver the required result when one, two, three or all four criteria aren’t populated or doesn’t match with any of the fields in the respective RANGE CRITERIA.

Can anyone help me?

Thanks Kaper!