I know some report showed H3K4me3 is found in the TSS of actively transcribing genes
But is H3K4me3 always associated with gene activation??
For example, can there be situation when the promoter of a gene has high H3K4me3, but the gene is downregulated due to other reasons (other mechanisms, like DNA methylation, transcription factors expression, insulators, etc)??
Well, from the 2007 Nature paper from Brad Bernstein's and Eric Lander's groups (Nature (448) pg 553-560), you can see that a number of promoters (at least in undifferentiated cells) have both the H3K4m3 and H3K27m3 mark and those promoters tend to be silent.
Also, genes that show an extreme degree of promoter proximal pausing (cMyc for instance, See papers from Groudine and Krumm) may have the H3K4m3 mark and Pol II recruitment but Pol II is paused just downstream of the TSS and very little if any continues to elongate through the rest of the gene.
In my own experience I have two genes which have a small degree of H3K4m3 but not detectable transcript (and barely detectable Pol II at the TSS).