Not true. Stealth fighters can stay stealthy with their radar on. They have Low Probability Intercept modes that keeps the aircraft pretty silent still even with radar on. It degrades stealth but it's not automatic detection.
LPI radars can still be detected. Similar to physical stealth, LPI radars decrease the range and probability of detection and increase the time needed for both detection and tracking. However, the longer duration of time a LPI radar emits, the higher the probability it will be detected and located. ESM systems have not stood at a standstill while radars evolved: modern ESM systems have adapted to counter the threat posed by LPI radars (via increased sensitivity and more advanced signals processing algorithms). Even with LPI mode, stealth fighters do not operate with their radars constantly on (they are used intermittently).
