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Generative Adversarial Networks (GANs) have proven to be a preferred method of synthesizing fake images of objects, such as faces, animals, and automobiles. It is not surprising these models can also generate ISO-compliant, yet synthetic iris images, which can be used to augment training data for iris matchers and liveness detectors. In this work, we trained one of the most recent GAN models (StyleGAN3) to generate fake iris images with two primary goals: (i) to understand the GAN's ability to produce "never-before-seen" irises, and (ii) to investigate the phenomenon of identity leakage as a function of the GAN's training time. Previous work has shown that personal biometric data can inadvertently flow from training data into synthetic samples, raising a privacy concern for subjects who accidentally appear in the training dataset. This paper presents analysis for three different iris matchers at varying points in the GAN training process to diagnose where and when authentic training samples are in jeopardy of leaking through the generative process. Our results show that while most synthetic samples do not show signs of identity leakage, a handful of generated samples match authentic (training) samples nearly perfectly, with consensus across all matchers. In order to prioritize privacy, security, and trust in the machine learning model development process, the research community must strike a delicate balance between the benefits of using synthetic data and the corresponding threats against privacy from potential identity leakage.