The correct answers are besmirch and tarnish.

Breakdown of the Blank

The sentence states that the mayor has "assiduously cultivating an image of integrity." The word "scandal" signals that the blank must be a negative action that would ruin this carefully crafted image.
- Besmirch: This means to make something dirty or to damage the reputation of someone or something in the eyes of others. This word fits perfectly, as a scandal would damage the mayor's reputation.
- Tarnish: This means to lose or cause to lose luster, especially as a result of exposure to air or moisture. Figuratively, it means to damage or spoil. In this context, it describes the process of a scandal spoiling the mayor's reputation.

The other options are incorrect:
- Bolster and solidify are the opposites of the intended meaning, as they mean to support or strengthen.
- Promulgate means to promote or make widely known, which is not the same as damaging a reputation.
- Mollify means to soothe the anger or anxiety of someone. This does not fit the context of a scandal affecting a reputation.