Litmus

Disclaimer: I swear that I did look up the word Litmus:

lit·mus Audio Help /ˈlɪtməs/ Pronunciation Key - Show Spelled Pronunciation[lit-muhs] Pronunciation Key - Show IPA Pronunciation
–noun
a blue coloring matter obtained from certain lichens, esp. Roccella tinctoria. In alkaline solution litmus turns blue, in acid solution, red: widely used as a chemical indicator.

However, the meaning I found does not fit this sentence:
"I'll give a litmus test of sincerity"

The problem is that you looked up the wrong thing. "litmus test" is a compound word.

litmus test
n.

1. A test for chemical acidity or basicity using litmus paper.
2. A test that uses a single indicator to prompt a decision: “The word ‘hopefully’ has become the litmus test to determine whether one is a language snob or a language slob” (William Safire).

Gotcha.
I found a more fitting meaning to the context of my original sentence thu:

a crucial and revealing test in which there is one decisive factor.