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It depends on whether the charges are free to move. There are two ways in which charge can move through a substance.

Ionic substances are made of positive and negative ions. In a solid they have fixed positions, but in liquids and gases they can move around. Ions are atoms (or groups of atoms) which have gained or lost electrons. How tightly their electrons are held has no effect here as it's the whole ion which moves.

The second situation where charges can move is when electrons are in delocalized bonds. In most substances the electrons are in orbitals associated with one atom or a small group of atoms. In metals, some electrons are in large orbitals which are shared by all the atoms, so the electrons (which are charged) can move freely through the whole metal. In this situation, the delocalized electrons can be thought of as not held by individual atoms, but they are still held by the assemblage of atoms. So the assertion in the question has elements of being right, but it's not the degree to which the electrons are held, but whether or not they are delocalized.

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Q: Does whether or not charges will move in a material depend partly on how tightly they are held in the atoms of the material?
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