If the director of building inspections shall be of opinion at any time and on any
reasonable grounds that the solvency of or financial responsibility on any bond of
any contractor is or may be insufficient or materially reduced by actual or possible
claims, then the director may make written demand on such contractor for an additional
bond which shall be of corresponding amount, form and sufficiency, and thereupon such
contractor shall immediately furnish such additional bond. The previous bond shall
not be held to cover any work not actually begun as aforesaid, before the date of
such new bond, but all work begun after such date shall be covered by such new bond.