May 9th is "Confession Day," a holiday that originates from the pun on the pronunciation of "go (5) ku (9) haku (confession)."
In other words, this jar of stars is equivalent to a confession.
Fushimi Roku felt slightly pleased, thinking that once the timing was right, he could poke at Minamoto Tamako's little scheme, pretending nonchalantly to say "Oh, your birthday gift just so happens to be 590 stars, are you secretly in love with me?"…
At that time, Minamoto Tamako would surely panic, lose her composure, and if she couldn't provide a reasonable explanation, he could further press on Minamoto Tamako until she would collapse and shout: "I do like you, so what!"
And with that, it would be mission accomplished.
He even already had his response line thought out: "Well... since you said so, I'll reluctantly agree to it."
Not bad, not bad.