  In this paper we ask when it is possible to transform a given sequence into a
frame or a lower semi frame by multiplying the elements by numbers. In other
words, we ask when a given sequence is a weighted frame or a weighted lower
semi frame and for each case we formulate a conjecture. We determine several
conditions under which these conjectures are true. Finally, we prove an
equivalence between two older conjectures, the first one being that any
unconditionally convergent multiplier can be written as a multiplier of Bessel
sequences by shifting of weights, and the second one that every unconditionally
convergent multiplier which is invertible can be written as a multiplier of
frames by shifting of weights. We also show that these conjectures are also
related to one of the newly posed conjectures.
