Good guess.
/ A / _ _ _ A / _ _ / _ _ / _ _ _ A _ _ _ / A / _ _ _ _ _ _ / _ _ / _ _ _ / _ A _ _ _ _ / .
Correct guesses so far: [‘A’]
Incorrect guesses so far: none
0 Strikes
I would like to guess O now, please.
Sorry, no O.
/ A / _ _ _ A / _ _ / _ _ / _ _ _ A _ _ _ / A / _ _ _ _ _ _ / _ _ / _ _ _ / _ A _ _ _ _ / .
Correct guesses so far: [‘A’]
Incorrect guesses so far: [‘O’]
Strikes: 1
There is a game where players are supposed to guess a 4-figure number, say 9753.
One can do this through successive strategic guesses, like 1234.
If I guess 1234 when the number is 9753, and consequently get only 1 figure (partially) correct, on my next guess I will try to get 0 figures correct so that I can eliminate all 4 figures from the total guess pool of 10 figures (which will allow me to focus on the only 6 remaining).
It’s a fun game, but I’m not sure my explanation is clear.
Anyway. Thank you. Now I would like to guess I (code word: India).
Yes, there are a few I’s.
/ A / _ _ _ A / _ _ / I _ / _ _ _ A _ _ _ / A / _ _ I _ _ _ / I _ / _ _ _ / _ A _ _ I _ / .
Correct guesses so far: [‘A’, ‘I’]
Incorrect guesses so far: [‘O’]
Strikes: 1
The game you describe is similar to a game I used to play called Mastermind, except with numbers instead of colored pegs.
Yes, it is based on the game I mentioned - I can see the English version is named Bulls and Cows.
I wonder if it would make a fun game on this board. 
Back to our game here. I would like to guess E now.
There are a couple of E’s.
/ A / _ E _ A / _ _ / I _ / _ _ _ A _ _ _ / A / _ _ I _ _ _ / I _ / _ _ E / _ A _ _ I _ / .
Correct guesses so far: [‘A’, ‘E’, ‘I’]
Incorrect guesses so far: [‘O’]
Strikes: 1
I will guess T now, please. 
Good guess. There are three T’s.
/ A / _ E _ A / _ _ / I _ / _ _ _ A _ _ _ / A / _ _ I T _ _ / I _ / T _ E / _ A T _ I _ / .
Correct guesses so far: [‘A’, ‘E’, ‘I’, ‘T’]
Incorrect guesses so far: [‘O’]
Strikes: 1
Thank you. I will pick one more letter before I take a break.
I would like to guess H.
Correct. There are two H’s.
/ A / _ E _ A / _ _ / I _ / _ _ _ A _ _ _ / A / _ _ I T _ H / I _ / T H E / _ A T _ I _ / .
Correct guesses so far: [‘A’, ‘E’, ‘H’, ‘I’, ‘T’]
Incorrect guesses so far: [‘O’]
Strikes: 1
Folks, keep the guesses coming, but I won’t be able to respond to them for the next six hours or so.
“S” please.
Here are the game rules again:
GAME RULES
It is a sentence hangman.
The original player posts the structure of the sentence.
The letters are not revealed since they are supposed to be found out by the rest of the players.
Players make their guesses.
Only letter guesses are admitted.
Sentence or word guesses are not allowed.
One guess per post.
The original player regularly updates the sentence by revealing the letters that have been guessed correctly.
Incorrect guesses are mentioned and tallied.
When five incorrect guesses have been accumulated, the original player wins and gets to post the next sentence.
If all the letters in the sentence have been guessed correctly before accumulating five incorrect guesses, the player who has made the last correct guess wins and gets to post the next sentence, thus becoming the original player for the next round.
I like your rigorous style. 
To be rigorous, I should bring Locrian’s guess here as well:
“S” please.
Yess, there are essess.
/ A / _ E _ A / _ _ / I S / _ S _ A _ _ _ / A / _ _ I T _ H / I _ / T H E / _ A T _ I _ / .
Correct guesses so far: [‘A’, ‘E’, ‘H’, ‘I’, ‘S’, ‘T’]
Incorrect guesses so far: [‘O’]
Strikes: 1
I’d like to guess N now, please.
There is an N, but only one.
/ A / _ E _ A / _ _ / I S / _ S _ A _ _ _ / A / _ _ I T _ H / I N / T H E / _ A T _ I _ / .
Correct guesses so far: [‘A’, ‘E’, ‘H’, ‘I’, ‘N’, ‘S’, ‘T’]
Incorrect guesses so far: [‘O’]
Strikes: 1