The 290 theorem states
If a positive-definite quadratic form with integer coefficients represents the twenty-nine integers $1$, $2$, $3$, $5$, $6$, $7$, $10$, $13$, $14$, $15$, $17$, $19$, $21$, $22$, $23$, $26$, $29$, $30$, $31$, $34$, $35$, $37$, $42$, $58$, $93$, $110$, $145$, $203$, and $290$, then it represents all positive integers.
Assume that I have such a form, e.g., $$ax^2+by^2+cz^2+dw^2\!,$$ which represents all 29 “critical integers”, so that by the theorem the form represents all positive integers.
Now assume further that I can prove one or more conditions on how the form actually represents one or more of the “critical integers“, e.g., say I can prove that $110$ can only be represented by the form if $x+y-z=1$.
QUESTION: Can it be shown that there exist an infinite number of positive integers subject to the extra condition(s)? Put another way, can the 290 Theorem can be refined/sharpened to show that any conditions which are required to represent all 29 of the “critical integers” are also required to represent all positive integers?