This article presents a refined notion of incompatible JPEG images for a quality factor of 100. It can be used to detect the presence of steganographic schemes embedding in DCT coefficients. We show that, within the JPEG pipeline, the combination of the DCT transform with the quantization function can map several distinct blocks in the pixel domain to the same block in the DCT domain. However, not every DCT block can be obtained: we call those blocks incompatible. In particular, incompatibility can happen when DCT coefficients are manually modified to embed a message. We show that the problem of distinguishing compatible blocks from incompatible ones is an inverse problem with or without solution and we propose two different methods to solve it. The first one is heuristic-based, fast to find a solution if it exists. The second is formulated as an Integer Linear Programming problem and can detect incompatible blocks only for a specific DCT transform in a reasonable amount of time. We show that the probability for a block to become incompatible only relies on the number of modifications. Finally, using the heuristic algorithm we can derive a Likelihood Ratio Test depending on the number of compatible blocks per image to perform steganalysis. We simulate the result of this test and show that it outperforms a deep learning detector e-SRNet for every payload between 0.001 and 0.01 bpp by using only 10% of the blocks from 256x256 images. A Selection-Channel-Aware version of the test is even more powerful and outperforms e-SRNet while using only 1% of the blocks.